Explicit Global Minimization of the Symmetrized Euclidean Distance by a Characterization of Real Matrices with Symmetric Square

We determine the optimal orthogonal matrices $R \in {{O}}(n)$ which minimize the symmetrized Euclidean distance $W\colon {{O}}(n) \to \Bbb{R}, \; W(R\,;D) \;:=\; \vert\vert{{sym}(R D - \mathbbm{1})}\vert\vert^2\,,$ where $\mathbb{1}$ denotes the identity matrix and ${sym}(X)=\frac{1}{2}(X + X^T)$ is the symmetric part of $X$, for a given positive definite diagonal matrix $D = \operatorname{diag}(d_1, \ldots , d_n)$ with distinct entries $d_1>d_2> \cdots > d_n > 0$. The number of critical points depends on $D$ and can grow faster than exponential in $n$. In the process, we prove and use a novel result of independent interest: every real matrix whose square is symmetric can be expressed as a block-diagonal matrix composed of blocks of size at most two by a suitable orthonormal change of basis.